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Fast exchange between the two peaks

Location: tutorial_simulations/Spectral_patterns/2/

Here we will analyze all the models/conditions that give rise to a fast-exchange shifting resonance like the one depicted below.

We will identify the distinct features of the line shapes originating from different models, which may helpful in establishing the 'dynamic identity' of the system. We will analyze peak positions, line widths and we will use fitting with the 2-site exchange model as a diagnostic tool.

Below I will do simulations for all the models and perform a side-by-side comparison at the end.

Contents

 

 

 

True 2-site exchange: U-model

 

Titration

The simple 2-site exchange without any coupled processes is the number one candidate to account for the fast exchange line shape. Let's simulate the line shape and fit it with the two-site exchange model. We expect to have a very good fit!

We make sure in the simulation to choose the spectral window wide enough so the tails of the peaks are included - to enable subsequent integration and fitting.

Simulate setup_1 U_Awf

Results: U_Awf/

Calculated populations of species

ITC profile 

  ITC curve is symmetrical around 1 (as expected if overall dilution of the receptor is minimal through the titration range).  

Simulated spectra
bottom->top increasing L/R

Chemical shift change upon titration

Line width change upon titration
(may be detected inaccurately, check against actual spectra)
Line shapes shift gradually with some broadening in the middle of the series. Chemical shift change is linear accurately following RL population change. Line width peaks at about 0.35. This is characteristic for the U model.

Fitting with the U model:

 

Optimum norm: 5.49e-03
[1] Kd: 5.99095e-07 +/- 1.8e-08
[2] Koff: 1.07488e+03 +/- 1.7e+01
[3] Scale Factor: 2.04922e+00 +/- 4.4e-03

  

 

 

Test chemical shift dependence of the final peak upon dilution

Simulate U_setup_conc_dep_final U_Awf

 

Conclusion:

20x dilution of 2.5:1=L:R mixture does NOT influence the complex peak position.

 

 

 

 

 

 

 

Conclusions

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U-R model with A and B in fast exchange

 

Test chemical shift dependence of the final peak upon dilution

Simulate U_R_setup_conc_dep_final U_R_Af_Bf

Conclusion:
Due to weakened affinity, dilution at 2.5 L/R causes some complex dissociation and a slight peak shift.

 

Titration

Run two versions with opposite and same signs of ddH in A and B transitions (ddH=dH(final)-dH(initial)):

Simulate U_R_setup U_R_Af_Bf

Simulate U_R_setup U_R_Af_Bf_inv_dH_B

Location: U_R/

 

Calculated populations of species

ITC profile assuming opposite signs of dH in A and B

ITC profile assuming the same sign of dH in A and B
The receptor population is split with a constant ratio across the titration range. R->R* and R->RL have the opposite signs of heat effects so R*->R->RL results in increased heat release. However, the shape of the curve stays symmetrical because there is no concentration dependence for R*->R process (to be analyzed in actual fitting). When signs are the same, R->R* reduces heat effect of R->RL proporionally across the entire range of L/R. ITC profile remains symmetrical.

Simulated spectra
bottom->top increasing L/R

Chemical shift change upon titration

Line width change upon titration
(may be detected inaccurately, check against actual spectra)
  Titration curve is linear accurately following the RL population change. Line width peaks at about 0.5 instead of 0.35, so poor fit with the U model may be predicted.

Fitting with the U model:

  
Note a remarkable discrepancy between the 2-site U-model and U_R data due to the mismatch of the line-broadening effect across the L/R range.  

 

 

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U-R2 model with A and B in fast exchange

The experiment #1 to identify this model is to detect the concentration dependence of peaks of the free receptor

Choose:

 

Simulate U_R2_conc_dep_setup U_R2_Af_Bf

Location: U_R2_conc_dep/

Comment:

Peaks shift towards the monomeric form upon 20x dilution.

 

Concentration dependence of the position of the final saturated complex at L/R =2.5

Choose:

 

Simulate U_R2_setup_conc_dep_final U_R2_Af_Bf

Conclusion:

The final complex is stable against 20x dilution at L/R=2.5 because the binding affinity is NOT weakened when [R] is tiny.

 

Titration

Run two versions with opposite and same signs of ddH in A and B transitions (ddH=dH(final)-dH(initial)):

Simulate U_R2_setup U_R2_Af_Bf

Simulate U_R2_setup U_R2_Af_Bf_inv_dH_B

Location: U_R2/

Calculated populations of species

Comment:

Populations of the receptor species are significantly shifted towards dimer, but the ratio of dimer/monomer rapidly decreases with the titration progress.

ITC profile assuming opposite signs of dH in A and B

Comment:

Titration curver eveals larger heat release in the beginning of titration than in the end, because of the non-linear contribution from R->RR process, making the beginning of the transition curve less sharp and rendering the profile slightly asymmetrical.

Stoichiometry as determined from the middle of the transition now deviates from 1 (about 1.05).

ITC profile assuming the same sign of dH in A and B

Comment:

Disassembly of RR now compensates for R-RL heat, more in the beginning than in the end of L/R range generating a negative pre-transition slope and making beginning of the transition sharper than the end.

Stoichiometry as determined from the middle of the transition also deviates from 1 (about 1.05).

Simulated spectra
bottom->top increasing L/R

Comment:

Line shapes look normal, in fast exchange.

Chemical shift change upon titration

Comment:

Note a non-linearity of the chemical shift plot. Poor fit with U model is expected.

Line width change upon titration
(may be detected inaccurately, check against actual spectra)

Comment:

Line broadening pattern is similar to the U model.

Fitting with the U model:

Comment:

Dramatic discrepancy between the U model fit and U_R2 data originates from the non-linear peak shifts in the beginning of titration.

 

 

Conclusion:

Combination of the concentration dependence and 2-site fit of a titration provide enough evidence for the U_R2 model

 

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U-L model with fast A

In this model kinetics of L<=>L* does not matter for the line shapes.

 

Concentration dependence of the saturated complex peak position at L/R=2.5

Choose:

 

Simulate U_L_setup_conc_dep_final U_L_Af_Bs

Conclusion:
Due to weakened affinity, dilution at 2.5 L/R causes some complex dissociation and a slight peak shift.

 

Titration

Simulate U_L_setup U_L_Af_Bs

Location: U_L/

Calculated populations of species

Comment:

The unbound ligand population is always split with the same ratio.

 

ITC profile 

Comment:

Titration curve is not accurate as heat effects of a titrant are not considered in this LineShapeKin Simulation version.

Simulated spectra
bottom->top increasing L/R

Comment:

Line shapes are normal.

 

Chemical shift change upon titration

Comment:

Chemical shift change is linear as in U model.

Line width change upon titration
(may be detected inaccurately, check against actual spectra)

Comment:

Line-broadening effect matches the one originating from the U model. Good fit is expected.

Fitting with the U model:

[1] Kd: 1.49764e-06 +/- 4.8e-08
[2] Koff: 1.03812e+03 +/- 3.0e+01
[3] Scale Factor: 2.04670e+00 +/- 8.2e-03

Comment:

The fit is remarkably good.

Ka=1/Kd= 0.67e+06 versus Ka=1e6 used in the simulation. Ka is supposed to be  scaled down by the competing formation of L*.

Koff is accurate: 1040+-30/s vs. 1000/s used in the simulation.

  

 

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U-L2 model with fast A

 

In this model kinetics of L<=>L* does not matter for the line shapes.

 

Dependence of the position of the final peak upon 20x dilution at L/R=2.5

 

Choose:

 

Simulate U_L2_setup_conc_dep_final U_L2_Af_Bs

Conclusion

Dilution dramatically increases amount of the free ligand, which opposes dissociation of the complex.

 

 

Titration

Simulate U_L2_setup U_L2_Af_Bs

Location: U_L2/

Calculated populations of species

Comment:

Population of RL grows slightly differently than in U_L and U models after the saturation point because as free L concentration increases it is progressively withdrawn from the reaction to form non-binding L2 species. Thus, after equivalence point it becomes increasingly difficult to reach saturation (as if we were adding less  and less ligand amounts with every next addition).

ITC profile assuming opposite signs of dH in A and B

Comment:

ITC profile assuming the same sign of dH in A and B

 

Comment:

Simulated spectra
bottom->top increasing L/R

Comment:

 

Chemical shift change upon titration

Comment:

Line width change upon titration
(may be detected inaccurately, check against actual spectra)

Comment:

Fitting with the U model:

Due to decreased apparent affinity the titration is not complete, so the final peak position is a poor estimate of the pure RL chemical shift. We need to fit dw(B).

1. Fit with fixed dw(B) to establish dw(B) estimate and the Scale Factor.

1.Matlab_fitting.Koff_SF/


2. Use fitted Scale Factor and dw(B) and fit with the variable dw(B):

4.Matlab_fitting.Koff_Kd_w0B.fixed_SF/

Fit is much better now.

[1] Kd: 2.64132e-06 +/- 1.8e-07
[2] Koff: 1.08698e+03 +/- 3.0e+01
[3] w0_B: 3.97157e+01 +/- 3.8e-02

Ka=1/Kd = 0.37e6  vs. 1e6 used for simulation

Koff=1090 /s - close to 1000/s used.

 

Comment:

Line shapes contain little information about the mechanism.

The major effect of dimerization in the free ligand, competing with binding, is reduction in the apparent binding affinity detected by line shape fitting.

  

 

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U-RL model with both A and B in fast exchange

 

 

Concentration dependence of the final peak's chemical shift on 20x dilution at L/R=2.5

LRratio 2.501 2.502 2.503 2.504 2.505

Rtotal 1e-3 0.5e-3 0.25e-3 0.125e-3 0.06e-3

Simulate U_RL_setup_conc_dep_final U_RL_Af_Bf

Conclusion

Isomerization of the complex makes binding tighter opposing dissociation upon dilution

 

 

Titration

Run two versions with the opposite and same signs of ddH in A and B transitions (ddH=dH(final)-dH(initial)):

Simulate U_RL_setup U_RL_Af_Bf

Simulate U_RL_setup U_RL_Af_Bf_inv_dH_B

Location: U_RL/

Calculated populations of species

Comment:

Population of the complex is always split between two isomers with the same ratio.

 

ITC profile assuming opposite signs of ddH in A and B

Comment:

Formation of R*L compensates some of the binding heat resulting in smaller signal than in the case of the same signs (right).

ITC profile assuming the same sign of ddH in A and B

Comment:

Simulated spectra
bottom->top increasing L/R

Comment:

Line shapes look pretty normal

 

Chemical shift change upon titration

Comment:

Line width change upon titration
(may be detected inaccurately, check against actual spectra)

Comment:

Line-broadening peaks at 0.4 indicating that a good fit with 2-site model is expected.

Fitting with the U model:

[1] Kd: 1.41354e-07 +/- 8.5e-09
[2] Koff: 1.43959e+02 +/- 1.0e+00

Comment:

The fit is relatively good, with a slight mismatch due to line width differences.

Remarkable that Ka=1/Kd= 7e+06, which is 7x larger than 1e6 used in the simulation.

In opposite, Koff=150/s, much slower than the k2(A)=1000/s and k2(B)=1000/s used in the simulation.

 

  

 

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U-R2L2 model with both A and B in fast exchange

 

The experiment #1 to identify this model is to detect the concentration dependence of peaks of the bound receptor

Choose:

Simulate U_R2L2_conc_dep_setup U_R2L2_Af_Bf

Location: U_R2L2_conc_dep/

Manually rescale X-axis on all graphs:

Comment:

Peak of the saturated receptor shifts significantly upon 20x dilution.

 

 

Simulate titrations

Run two versions with the opposite and same signs of ddH in A and B transitions (ddH=dH(final)-dH(initial)):

Simulate U_R2L2_setup U_R2L2_Af_Bf

Simulate U_R2L2_setup U_R2L2_Af_Bf_inv_dH

 

Calculated populations of species

Comment:

Bound receptor population is split between the monomeric and dimeric species with dimer/monomer ratio progressively increasing with L/R.

ITC profile assuming opposite signs of ddH in A and B

Comment:

In the beginning when little dimer is formed the heat effect is maximal due to formation of RL. As concentration of RL rises and it begins to dimerize the opposite sign of dimerization heat leads to a reduction in total heat release due to titration.

ITC profile assuming the same sign of ddH in A and B

Comment:

Same but now a dimer formation progressively contributes more and more heat to the titration as saturation level increases.

Simulated spectra
bottom->top increasing L/R

Comment:

Line shapes look pretty normal.

 

Chemical shift change upon titration

Comment:

Non-linear chemical shift change will certainly make U-model fit poor.

Line width change upon titration
(may be detected inaccurately, check against actual spectra)

Comment:

Line-broadening effect is maximum at 0.6, which will make it incompatible with the U model (maximum at 0.35).

Fitting with the U model:

Comment:

As expected from chemical shift change and the line-broadening distribution, the U-model fit fails miserably.

 

  

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Conclusions

 

See side-by-side comparison here

 

 

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